Let's walk through this step-by-step, assuming the transformations have been applied and considering their implications:
1. Angle Preservation under Rotation and Dilation:
- A rotation, such as \( D_{0,0,75}(x, y) \), changes the orientation of the triangle but does not change the size or angle measures. Angles remain congruent.
- Similarly, a dilation, such as \( D_{0,2}(x,y) \), scales the triangle but does not alter the angles within the triangle. The ratio of corresponding sides changes, but the angles remain unchanged.
Therefore, \(\angle M \cong \angle M'\) must be true.
2. Equality of the Triangles:
- The rotation and dilation together would change both the orientation and the dimensions of \(\triangle LMN\).
- Specifically, dilation by a factor of 2 enlarges the triangle by twice its size, making \(\triangle LMN \neq \triangle L' M' N'\).
Therefore, \(\triangle LMN \neq \triangle L' M' N'\) must be true.
3. Similarity of the Triangles:
- Since dilation changes the side lengths proportionally and rotation changes orientation without affecting side lengths and angles, the resulting triangle will be similar to the original triangle.
Thus, \(\triangle LMN \sim \triangle L' M' N'\) must be true.
4. Coordinates of Vertices:
- Given the coordinates obtained from the transformations and verifying their truthfulness:
- The coordinates of vertex \(L''\) are \((-3, 1.5)\).
- The coordinates of vertex \(N'\) are \( (3, -1.5) \).
- The coordinates of vertex \(M'\) are \( (1.5, -1.5) \).
These statements and coordinates allow us to conclude:
- \(\angle M \cong \angle M'\)
- \(\triangle LMN \neq \triangle L' M' N'\)
- \(\triangle LMN \sim \triangle L' M' N'\)
- The coordinates of vertex \(L''\) are \((-3, 1.5)\)
- The coordinates of vertex \(N'\) are \((3, -1.5)\)
- The coordinates of vertex \(M'\) are \((1.5, -1.5)\)
Therefore, the correct statements to check are:
- \(\angle M \cong \angle M'\)
- \(\triangle LMN \neq \triangle L' M' N'\)
- \(\triangle LMN \sim \triangle L' M' N'\)
- The coordinates of vertex \(L''\) are \((-3, 1.5)\)
- The coordinates of vertex \(N'\) are \((3, -1.5)\)
- The coordinates of vertex [tex]\(M'\)[/tex] are [tex]\((1.5, -1.5)\)[/tex]
